Based on a recent characterization of the strong maximum principle, [3], this paper gives some periodic parabolic counterparts of some of the results of Chapters 8 and 9 of J. L´opez-G´omez [22]. Among them count some pivotal monotonicity properties of the principal eigenvalue σ[P+V,B,QT ], as well as its concavity with respect to the periodic potential V through a point-wise periodic-parabolic Donsker–Varadhan min-max characterization. Finally, based on these findings, this paper sharpens, substantially, some classical results of A. Beltramo and P. Hess [4], K. J. Brown and S. S. Lin [6], and P. Hess [14] on the existence and uniqueness of principal eigenvalues for weighted boundary value problems.